Mathematics UPSC PYQ 2015

Give an example of a ring having identity but a subring of this having a different identity.

A body moving under SHM has an amplitude 'a' and time period 'T'. If the velocity is trebled, when the distance from mean position is 2/3 a, the period being unaltered, find the new amplitude.

Test the convergence and absolute convergence of the series sum_{n=1}^{infty} (-1)^{n+1} \frac{n}{n^2 + 1}

Find the dimension of the subspace of R^4, spanned by the set {(1, 0, 0, 0), (0, 1, 0, 0), (1, 2, 0, 1), (0, 0, 0, 1)} Hence find its basis.

If matrix A = [ [1 0 0], [1 0 1], [0 1 0] ] then find A^30.

The set of numbers of the form b\sqrt{2} with b rational

Find the constant a so that (x + y)^a is the Integrating factor of (4x^2 + 2xy + 6y)dx + (2x^2 + 9y + 3x)dy = 0 and hence solve the differential equation.

Is the function f(x) = { 1/n, 1/(n+1) < x <= 1/n; 0, x = 0 } Riemann integrable? If yes, obtain the value of integral_{0}^{1} f(x) dx.

Show that the function v(x, y) = ln(x^2 + y^2) + x + y is harmonic. Find its conjugate harmonic function u(x, y). Also, find the corresponding analytic function f(z) = u + iv in terms of z.

Evaluate line_integral_C (e^{-x} (sin y dx + cos y dy)), where C is the rectangle with vertices (0, 0), (pi, 0), (pi, pi/2), (0, pi/2).

Which point of the sphere x2 + y2 + z2 = 1 is at the maximum distance from the point (2, 1, 3) ?

Do the following sets form integral domains with respect to ordinary addition and multiplication? If so, state if they are fields : (i) The set of numbers of the form b\sqrt{2} with b rational (ii) The set of even integers (iii) The set of positive integers

Evaluate the integral \iint_{R} (x - y)^2 \cos^2(x + y) dx dy where R is the rhombus with successive vertices as (\pi, 0) (2\pi, \pi) (\pi, 2\pi) (0, \pi).

Find the Lagrange interpolating polynomial that fits the following data : x : -1, 2, 3, 4 f(x) : -1, 11, 31, 69 Find f(1.5).

Which point of the sphere x^2 + y^2 + z^2 = 1 is at the maximum distance from the point (2, 1, 3)?

Obtain Laplace Inverse transform of { \ln(1 + \frac{1}{s}) + \frac{s}{s^2 + 25} e^{-\pi s} }

State Cauchy's residue theorem. Using it, evaluate the integral integral_{C} \frac{e^z + 1}{z(z+1)(z-i)^2} dz; C: |z| = 2

Solve (D^2 + DD' - 2D'^2)u = e^{x+y}, where D = \frac{\partial}{\partial x} and D' = \frac{\partial}{\partial y}.

Consider a uniform flow U_0 in the positive x-direction. A cylinder of radius a is located at the origin. Find the stream function and the velocity potential. Find also the stagnation points.

Find an equivalent Lagrangian that is not explicitly dependent on time.

A rod of 8 kg is movable in a vertical plane about a hinge at one end, another end is fastened a weight equal to half of the rod, this end is fastened by a string of length l to a point b above the hinge vertically. Obtain the tension in the string.

Reduce the second-order partial differential equation x^2 \frac{\partial^2 u}{\partial x^2} - 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2} + x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 0 into canonical form. Hence, find its general solution.

A body moving under SHM has an amplitude 'a' and time period 'T'. If the velocity is trebled, when the distance from mean position is \frac{2}{3}a, the period being unaltered, find the new amplitude.

Verify if the lines : (x - a + d) / (alpha - delta) = (y - a) / (alpha) = (z - a - d) / (alpha + delta) and (x - b + c) / (beta - gamma) = (y - b) / (beta) = (z - b - c) / (beta + gamma) are coplanar. If yes, then find the equation of the plane in which they lie.

The set of even integers

Obtain the equation of the plane passing through the points (2, 3, 1) and (4, -5, 3) parallel to x-axis.

For what positive value of 'a', the plane ax - 2y + z + 12 = 0 touches the sphere x^2 + y^2 + z^2 - 2x - 4y + 2z - 3 = 0 and hence find the point of contact.

Solve the differential equation : (2xy^4 + 2xy^3 + y)dx + (x^2y^4 - x^2y - 3x)dy = 0.

The set of positive integers

Evaluate double_integral_R sqrt(y - x^2) dx dy where R = [-1, 1; 0, 2].

Solve the plane pendulum problem using the Hamiltonian approach and show that H is a constant of motion.

A vector field is given by F_vector = (x^2 + xy^2)i_vector + (y^2 + x^2 y)j_vector Verify that the field F_vector is irrotational or not. Find the scalar potential.

Find all possible Taylor's and Laurent's series expansions of the function f(z) = \frac{2z - 3}{z^2 - 3z + 2} about the point z = 0.

Test the series of functions \sum_{n=1}^{\infty} \frac{nx}{(1+n^2x^2)} for uniform convergence.

Calculate the moment of inertia of a solid uniform hemisphere x^2 + y^2 + z^2 = a^2, z >= 0 with mass m about the OZ-axis.

Let V = R^3 and T in A(V), for all a_i in A(V), be defined by T(a_1, a_2, a_3) = (2a_1 + 5a_2 + a_3, -3a_1 + a_2 - a_3, -a_1 + 2a_2 + 3a_3) What is the matrix T relative to the basis V_1 = (1, 0, 1) V_2 = (-1, 2, 1) V_3 = (3, -1, 1)?

Let V = R3 and T \in A(V), for all a_i \in A(V), be defined by T(a_1, a_2, a_3) = (2a_1 + 5a_2 + a_3, -3a_1 + a_2 - a_3, -a_1 + 2a_2 + 3a_3) What is the matrix T relative to the basis V1 = (1, 0, 1) V2 = (-1, 2, 1) V3 = (3, -1, 1) ?

Solve the following linear programming problem by the simplex method. Write its dual. Also, write the optimal solution of the dual from the optimal table of the given problem : Maximize Z = 2x_1 - 4x_2 + 5x_3 subject to x_1 + 4x_2 - 2x_3 \le 2 -x_1 + 2x_2 + 3x_3 \le 1 x_1, x_2, x_3 \ge 0

Evaluate \iint_{R} \sqrt{|y - x^2|} dx dy where R = [-1, 1 ; 0, 2].

Calculate the moment of inertia of a solid uniform hemisphere x^2 + y^2 + z^2 = a^2, z \ge 0 with mass m about the OZ-axis.

Verify if the lines : \frac{x - a + d}{\alpha - \delta} = \frac{y - a}{\alpha} = \frac{z - a - d}{\alpha + \delta} and \frac{x - b + c}{\beta - \gamma} = \frac{y - b}{\beta} = \frac{z - b - c}{\beta + \gamma} are coplanar. If yes, then find the equation of the plane in which they lie.

Solve the differential equation : x \cos x \frac{dy}{dx} + y(x \sin x + \cos x) = 1.

Obtain Laplace Inverse transform of {ln(1 + 1/s) + s/(s^2 + 25) e^{-rs}}.

Taking a group (e, a, b, c) of order 4, where e is the identity, construct composition tables showing that one is cyclic while the other is not.

A particle is projected from the base of a hill whose slope is that of a right circular cone, whose axis is vertical. The projectile grazes the vertex and strikes the hill again at a point on the base. If the semivertical angle of the cone is 30 degrees, h is height, determine the initial velocity u of the projection and its angle of projection.

Solve x^4 d^4y/dx^4 + 6x^3 d^3y/dx^3 + 4x^2 d^2y/dx^2 - 2x dy/dx - 4y = x^2 + 2 cos(log_e x).

Find the dimension of the subspace of R4, spanned by the set {(1, 0, 0, 0), (0, 1, 0, 0), (1, 2, 0, 1), (0, 0, 0, 1)} Hence find its basis.

Without solving the problem, show that it has an optimal solution. Which of the basic feasible solution(s) is/are optimal?

Find the angle between the surfaces x^2 + y^2 + z^2 - 9 = 0 and z = x^2 + y^2 - 3 at (2, -1, 2).

Using Laplace transform, solve y'' + y = t, y(0) = 1, y'(0) = -2.

Find the principal (or canonical) disjunctive normal form in three variables p, q, r for the Boolean expression ((p \land q) \to r) \lor ((p \land q) \to \neg r). Is the given Boolean expression a contradiction or a tautology?

Solve the differential equation x = py - p^2 where p = dy/dx.

A vector field is given by F = (x^2 + xy^2)i + (y^2 + x^2y)j Verify that the field F is irrotational or not. Find the scalar potential.

Find the absolute maximum and minimum values of the function f(x, y) = x^2 + 3y^2 - y over the region x^2 + 2y^2 \le 1.

Evaluate the following integral : int_{pi/6}^{pi/3} (sqrt(sin x) / (sqrt(sin x) + sqrt(cos x))) dx.

A mass starts from rest at a distance 'a' from the centre of force which attracts inversely as the distance. Find the time of arriving at the centre.

Solve the following assignment problem to maximize the sales : Territories (क्षेत्र) I II III IV V A 3 4 5 6 7 B 4 15 13 7 6 Salesmen (विक्रेता) C 6 13 12 5 11 D 7 12 15 8 5 E 8 13 10 6 9

Find a Lagrangian corresponding to this Hamiltonian.

Evaluate the following limit : Lt_{x->a} (2 - x/a)^{tan(pi x)/(2a)}

Solve the differential equation x = py - p^2 where p = \frac{dy}{dx}.

For the function f(x, y) = \begin{cases} \frac{x^2 - x\sqrt{y}}{x^2 + y}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases} Examine the continuity and differentiability.

Using the definition, find its all basic solutions. Which of these are degenerate basic feasible solutions and which are non-degenerate basic feasible solutions?

A rod of 8 kg is movable in a vertical plane about a hinge at one end, another end is fastened a weight equal to half of the rod, this end is fastened by a string of length l to a point at a height b above the hinge vertically. Obtain the tension in the string.

Reduce the following matrix to row echelon form and hence find its rank : [ 1 2 3 4 ] [ 2 1 4 5 ] [ 1 5 5 7 ] [ 8 1 14 17 ]

If 6x = 3y = 2z represents one of the three mutually perpendicular generators of the cone 5yz - 8zx - 3xy = 0 then obtain the equations of the other two generators.

The vectors V1 = (1, 1, 2, 4), V2 = (2, -1, -5, 2), V3 = (1, -1, -4, 0) and V4 = (2, 1, 1, 6) are linearly independent. Is it true ? Justify your answer.

How many generators are there of the cyclic group G of order 8? Explain.

Find the solution of the initial-boundary value problem u_t - u_{xx} + u = 0, 0 < x < l, t > 0 u(0, t) = u(l, t) = 0, t >= 0 u(x, 0) = x(l - x), 0 < x < l

Evaluate \int_C e^{-x} (\sin y dx + \cos y dy), where C is the rectangle with vertices (0, 0), (\pi, 0), (\pi, \frac{\pi}{2}), (0, \frac{\pi}{2}).

A conical tent is of given capacity. For the least amount of Canvas required, for it, find the ratio of its height to the radius of its base.

For what positive value of a, the plane ax - 2y + z + 12 = 0 touches the sphere x2 + y2 + z2 - 2x - 4y + 2z - 3 = 0 and hence find the point of contact.

Find the value of \lambda and \mu so that the surfaces \lambda x^2 - \mu yz = (\lambda + 2)x and 4x^2y + z^3 = 4 may intersect orthogonally at (1, -1, 2).

Find the length of an endless chain which will hang over a circular pulley of radius 'a' so as to be in contact with the two-thirds of the circumference of the pulley.

Solve : x^4 \frac{d^4y}{dx^4} + 6x^3 \frac{d^3y}{dx^3} + 4x^2 \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} - 4y = x^2 + 2\cos(\log_e x).

The vectors V_1 = (1, 1, 2, 4), V_2 = (2, -1, -5, 2), V_3 = (1, -1, -4, 0) and V_4 = (2, 1, 1, 6) are linearly independent. Is it true ? Justify your answer.

A particle is projected from the base of a hill whose slope is that of a right circular cone, whose axis is vertical. The projectile grazes the vertex and strikes the hill again at a point on the base. If the semivertical angle of the cone is 30°, h is height, determine the initial velocity u of the projection and its angle of projection.

Find the solution of the system 10x_1 - 2x_2 - x_3 - x_4 = 3 -2x_1 + 10x_2 - x_3 - x_4 = 15 -x_1 - x_2 + 10x_3 - 2x_4 = 27 -x_1 - x_2 - 2x_3 + 10x_4 = -9 using Gauss-Seidel method (make four iterations).

Solve for the general solution p \cos(x+y) + q \sin(x+y) = z, where p = \frac{\partial z}{\partial x} and q = \frac{\partial z}{\partial y}.

In an axisymmetric motion, show that stream function exists due to equation of continuity. Express the velocity components in terms of the stream function. Find the equation satisfied by the stream function if the flow is irrotational.

Solve the differential equation : (2xy^4 e^y + 2xy^3 + y)dx + (x^2 y^4 e^y - x^2 y^2 - 3x)dy = 0.

Evaluate the following limit : Lt_{x \to a} (2 - \frac{x}{a})^{tan(\frac{\pi x}{2a})}

Solve the following linear programming problem by the simplex method. Write its dual. Also, write the optimal solution of the dual from the optimal table of the given problem : Maximize Z = 2x_1 - 4x_2 + 5x_3 subject to x_1 + 4x_2 - 2x_3 <= 2 -x_1 + 2x_2 + 3x_3 <= 1 x_1, x_2, x_3 >= 0

Two perpendicular tangent planes to the paraboloid x2 + y2 = 2z intersect in a straight line in the plane x = 0. Obtain the curve to which this straight line touches.

Consider the following linear programming problem : Maximize Z = x_1 + 2x_2 - 3x_3 + 4x_4 subject to x_1 + x_2 + 2x_3 + 3x_4 = 12 x_2 + 2x_3 + x_4 = 8 x_1, x_2, x_3, x_4 \ge 0 (i) Using the definition, find its all basic solutions. Which of these are degenerate basic feasible solutions and which are non-degenerate basic feasible solutions? (ii) Without solving the problem, show that it has an optimal solution. Which of the basic feasible solution(s) is/are optimal?

Two equal ladders of weight 4 kg each are placed so to lean at A against each other with their ends resting on a rough floor, given the coefficient of friction is mu. The ladders at A make an angle 60 degrees with each other. Find what weight on the top would cause them to slip.

Find the eigen values and eigen vectors of the matrix : [ [1 1 3], [1 5 1], [3 1 1] ]

Find the principal (or canonical) disjunctive normal form in three variables p, q, r for the Boolean expression ((p ^ q) -> r) v ((p ^ q) -> ~r). Is the given Boolean expression a contradiction or a tautology?

A particle moves in a plane under a force, towards a fixed centre, proportional to the distance. If the path of the particle has two apsidal distances a, b (a > b), then find the equation of the path.

Two perpendicular tangent planes to the paraboloid x^2 + y^2 = 2z intersect in a straight line in the plane x = 0. Obtain the curve to which this straight line touches.

If R is a ring with unit element 1 and \phi is a homomorphism of R onto R', prove that \phi(1) is the unit element of R'.

How many generators are the cyclic group of order 8? Explain.

Solve the partial differential equation (y^2 + z^2 - x^2)p - 2xyq + 2xz = 0 where p = \frac{\partial z}{\partial x} and q = \frac{\partial z}{\partial y}.

Find the absolute maximum and minimum values of the function f(x, y) = x^2 + 3y^2 - y over the region x^2 + 2y^2 <= 1.

State Cauchy's residue theorem. Using it, evaluate the integral \int_{C} \frac{e^z+1}{z(z+1)(z-i)^2} dz; C: |z|=2

Find the value of lambda and mu so that the surfaces lambda x^2 - muyz = (lambda + 2)x and 4x^2 y + z^3 = 4 may intersect orthogonally at (1, -1, 2).

A Hamiltonian of a system with one degree of freedom has the form H = \frac{p^2}{2\alpha} - bqpe^{-\alpha t} + \frac{b\alpha}{2}q^2 e^{-\alpha t}(\alpha + be^{-\alpha t}) + \frac{k}{2}q^2 where \alpha, b, k are constants, q is the generalized coordinate and p is the corresponding generalized momentum. (i) Find a Lagrangian corresponding to this Hamiltonian. (ii) Find an equivalent Lagrangian that is not explicitly dependent on time.

Test the series of functions sum_{n=1}^{infty} \frac{nx}{(1 + n^2 x^2)} for uniform convergence.

Evaluate the following integral : \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt[3]{\sin x}}{\sqrt[3]{\sin x} + \sqrt[3]{\cos x}} dx.

For the function f(x, y) = { (x^2 - x*sqrt(y))/(x^2 + y), (x, y) != (0, 0); 0, (x, y) = (0, 0) } Examine the continuity and differentiability.

Solve the initial value problem \frac{dy}{dx} = x(y - x), y(2) = 3 in the interval [2, 2.4] using the Runge-Kutta fourth-order method with step size h = 0.2.

If matrix A = [ [1, 0, 0], [1, 0, 1], [0, 1, 0] ] then find A30.

Evaluate the integral double_integral_R (x - y)^2 cos^2(x + y) dx dy where R is the rhombus with successive vertices as (pi, 0) (2pi, pi) (pi, 2pi) (0, pi).

Is the function f(x) = { 1/n, \frac{1}{n+1} < x \le \frac{1}{n}; 0, x = 0 } Riemann integrable? If yes, obtain the value of \int_{0}^{1} f(x) dx.

Find the solution of the initial-boundary value problem u_t - u_{xx} + u = 0, 0 < x < l, t > 0 u(0, t) = u(l, t) = 0, t \ge 0 u(x, 0) = x(l-x), 0 < x < l

Two equal ladders of weight 4 kg each are placed so as to lean at A against each other with their ends resting on a rough floor, given the coefficient of friction is \mu. The ladders at A make an angle 60° with each other. Find what weight on the top would cause them to slip.

Solve the differential equation : x cos x dy/dx + y(x sin x + cos x) = 1.